Abstract
The aim of this paper is to solve the open problem on product properties of digital covering maps raised from [5]. Namely, let us consider the digital images $X_1 {\subset}Z^{n_{0}}$ with $k_0-adjacency$, $Y_1{\subset}Z^{n_{1}}$ with $k_3-adjacency$, $X_2{\subset}Z^{n_{2}}$ with $k_2-adjacency$ and $Y_2{\subset}Z^{n_{3}}$ with $k_3-adjacency$. Then the reasonable $k_4-adjacency$ of the product image $X_1{\times}X_2$ is determined by the $k_0-$ and $k_2-adjacency$ and the suitable k_5-adjacency$ is assumed on $Y_1{\times}Y_2$ via the $k_1-$ and $k_3-adjacency$ [3] such that each of the projection maps is a digitally continuous map, e.g., $p_1\;:\;X_1{\times}X_2{\rightarrow}X_1$ is a digitally ($k_4,\;k_1$)-continuous map and so on. Let us assume $h_1\;:\;X_1{\rightarrow}Y_1$ to be a digital $(k_0, k_1)$-covering map and $h_2\;:\;X_2{\rightarrow}Y_2$ to be a digital $(k_2,\;k_3)$-covering map. Then we show that the product map $h_1{\times}h_2\;:\;X_1{\times}X_2{\rightarrow}Y_1{\times}Y_2$ need not be a digital $(k_4,k_5)$-covering map.